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            <h1 style="display: none">algorithm-线性基</h1>
            
              <p class="note note-info">
                
                  本文最后更新于：1 年前
                
              </p>
            
            <div class="markdown-body" id="post-body">
              <h2 id="前言"><a href="#前言" class="headerlink" title="前言"></a>前言</h2><blockquote>
<p>线性基是向量空间的一组基，通常可以解决有关异或的一些题目。──OI Wiki</p>
</blockquote>
<ul>
<li>另外，为了描述方便，我们不严谨的约定下文中出现的所有“<strong>子集</strong>”都非空</li>
</ul>
<a id="more"></a>
<h3 id="线性基的作用"><a href="#线性基的作用" class="headerlink" title="线性基的作用"></a>线性基的作用</h3><ul>
<li><p>求解一个数集中 集合内异或和最大/最小的子集中元素的异或和（即从集合中挑选任意多个数，求异或和的最值）</p>
</li>
<li><p>在线性基的帮助下，一个无序的异或和问题可以转化为贪心问题</p>
</li>
</ul>
<h3 id="线性基的性质"><a href="#线性基的性质" class="headerlink" title="线性基的性质"></a>线性基的性质</h3><p><del>虽然我不知道为什么所有有关线性基的文章里都会把这个放在前面，但是我还是放一下吧</del></p>
<ul>
<li>① 线性基所有子集的异或和形成的集合 与 原数集所有子集的异或和形成的集合 相同<ul>
<li>(线性基内所有的元素都是通过原数集中的元素做异或运算得到的)</li>
</ul>
</li>
<li>② 线性基是满足①性质的最小集合</li>
<li>③ 线性基不存在异或和为$0$的子集<ul>
<li>(线性基所有元素的得到不存在与$0$异或的过程)</li>
</ul>
</li>
<li>④ 线性基所有子集的异或和形成的集合为无重集合<ul>
<li>(本质上和性质③一样)</li>
</ul>
</li>
<li>⑤ 线性基内元素二进制下最高位都在不同的位置<ul>
<li>(因为线性基的实现就是以最高位位置为索引存的一维数组)</li>
</ul>
</li>
</ul>
<h3 id="线性基的构造"><a href="#线性基的构造" class="headerlink" title="线性基的构造"></a>线性基的构造</h3><p>似乎所有题解都是在摆出一大堆说明后直接开始讲怎么构造(虽然似乎这样更有效率)，但是我其实一开始看得一头雾水也挺悲(虽然其实真的直接看会快很多)，但我还是决定把分析推导过程谈一谈(当然只是我自己的)。</p>
<p>考虑异或和最大问题</p>
<p>首先有关异或这个运算的特性，不再详细说明，提出关键词：可逆</p>
<p>想让一个数最大，在二进制下我们肯定希望它的最高位的位置能更大，一个最高位在第8位的肯定比最高位在第4位的大。</p>
<p>然后，比如说我们现在已经有一个二进制数$(10100)_2$，我们想让他更大，可以让它与$(1xxxxx)_2$异或，不管怎么样它都能变大。所以我们考虑能不能维护一个数组，下标是最高位在第$i+1$位的数。</p>
<p>考虑一个最高位位置各不相同的集合，可以直接按照位置从高到低贪心求解。</p>
<p>然而实际问题里最高位位置不一定都不同。所以我们的线性基不能直接简单存储。我们现在考虑最高位在同一个位置的两个数怎么处理。</p>
<p>比如有：$x=(1010)_2$，$y=(1001)_2$，它们的最高位都是第四位。考虑处理顺序，我们先将$x$塞入了$b[3]$（因为第四位实际上是用(1&lt;&lt;3)获得的，所以我们用$i-1$表示第$i$位），然后发现$y$的最高位也是4。如果这两个数都选择的话，就会导致第四位的$1$消失，所以我们考虑把$y =y\oplus x$，虽然$b[3]$存的是$x$，但是除了第四位的信息，都可以由新的$y$和$x$得到，这时候我们把$y$再塞入线性基就可以了。如果遇到冲突处理类似。</p>
<p>当我们在贪心处理的时候，如果$x$本来就比$y$大，那我们直接就不用$y$了。如果$y$比$x$大，那我们可以用$x\oplus y$来表示原来的$y$。</p>
<p>简单且不严谨的归纳一下，由于异或运算的可逆性，存$x，x\oplus  y$和存$x，y$是一样的，但是可以造成其中一个数最高位的变化。</p>
<p>简单的理解是，线性基维护的是能够对第$i$位产生贡献的元素中的其中一个，并通过之后的部分元素与第$i$位元素共同维护原集合中最高位在$i$的元素。</p>
<h3 id="线性基的实现"><a href="#线性基的实现" class="headerlink" title="线性基的实现"></a>线性基的实现</h3><p>对于一个数$x$：</p>
<ul>
<li>找到它最高位的位置$p$（满足$x \&amp; (1&lt;&lt;p)=1$的最大的$p$）</li>
<li>检查<code>b[p]</code>是否为空<ul>
<li>如果为空：令<code>b[p]</code>为$x$</li>
<li>如果非空：令$x$为$x\oplus b[p]$，重复跳回最初</li>
</ul>
</li>
</ul>
<p>最终得到线性基数组<code>b[n]</code></p>
<div class="hljs"><pre><code class="hljs cpp"><span class="hljs-comment">//find函数返回p</span>
rep(i,<span class="hljs-number">1</span>,n)&#123;
        <span class="hljs-keyword">while</span>(<span class="hljs-number">1</span>)&#123;
            <span class="hljs-keyword">int</span> p = find(a[i]);
            <span class="hljs-keyword">if</span>(!b[p])&#123;
                b[p] = a[i];
                <span class="hljs-keyword">break</span>;
            &#125; <span class="hljs-keyword">else</span> &#123;
                <span class="hljs-keyword">if</span>(p == <span class="hljs-number">0</span>) <span class="hljs-keyword">break</span>;
                a[i] ^= b[p];
            &#125;
        &#125; 
    &#125;</code></pre></div>
<h3 id="线性基的应用"><a href="#线性基的应用" class="headerlink" title="线性基的应用"></a>线性基的应用</h3><h4 id="求异或和最小"><a href="#求异或和最小" class="headerlink" title="求异或和最小"></a>求异或和最小</h4><p>直接返回<code>b[n]</code>中的最小元素（即从小往大遍历第一个非$0$元素）</p>
<h4 id="求异或和最大"><a href="#求异或和最大" class="headerlink" title="求异或和最大"></a>求异或和最大</h4><p>从大往小贪心，如果<code>b[i]</code>能使$ans$变大，那就异或<code>b[i]</code>，反之不异或。</p>
<h3 id="相关题目"><a href="#相关题目" class="headerlink" title="相关题目"></a>相关题目</h3><p><strong><a target="_blank" rel="noopener" href="https://www.luogu.com.cn/problem/P3812">P3812 【模板】线性基</a></strong></p>
<div class='spoiler collapsed'>
    <div class='spoiler-title'>
        Code
    </div>
    <div class='spoiler-content'>
        <div class="hljs"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;map&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;set&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cmath&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;queue&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;bitset&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;vector&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdio&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstring&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdlib&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;iostream&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;algorithm&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> rep(i,a,b) for(register int i = (a);i &lt;= (b);++i)</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> per(i,a,b) for(register int i = (a);i &gt;= (b);--i) </span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> mkp std::make_pair</span>
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ll;
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">unsigned</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ull;
<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">string</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cin</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cout</span>;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x,<span class="hljs-keyword">int</span> y)</span></span>&#123;<span class="hljs-keyword">return</span> x &lt; y;&#125;

<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> N = <span class="hljs-number">1e6</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> inf = <span class="hljs-number">1e9</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">double</span> eps = <span class="hljs-number">1e-7</span>;
<span class="hljs-keyword">int</span> _,n,m;
ll a[<span class="hljs-number">2</span>*N],b[<span class="hljs-number">110</span>];

<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">int</span> <span class="hljs-title">find</span><span class="hljs-params">(ll x)</span></span>&#123;
    <span class="hljs-keyword">int</span> p = <span class="hljs-number">55</span>;
    <span class="hljs-keyword">while</span>((<span class="hljs-number">1l</span>l&lt;&lt;p) &gt; x &amp;&amp; p &gt;= <span class="hljs-number">0</span>) --p;
    <span class="hljs-keyword">while</span>((<span class="hljs-number">1l</span>l&lt;&lt;p) &amp; x == <span class="hljs-number">0</span> &amp;&amp; p &gt;= <span class="hljs-number">0</span>) --p;
    <span class="hljs-keyword">return</span> p;
&#125;

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    <span class="hljs-built_in">std</span>::ios::sync_with_stdio(<span class="hljs-number">0</span>);<span class="hljs-built_in">cin</span>.tie(<span class="hljs-number">0</span>);<span class="hljs-built_in">cout</span>.tie(<span class="hljs-number">0</span>);
    <span class="hljs-comment">//freopen(&quot;in.in&quot;, &quot;r&quot;, stdin);</span>
    <span class="hljs-built_in">cin</span> &gt;&gt; n;
    rep(i,<span class="hljs-number">1</span>,n) <span class="hljs-built_in">cin</span> &gt;&gt; a[i];
    rep(i,<span class="hljs-number">1</span>,n)&#123;
        <span class="hljs-keyword">while</span>(<span class="hljs-number">1</span>)&#123;
            <span class="hljs-keyword">int</span> p = find(a[i]);
            <span class="hljs-keyword">if</span>(!b[p])&#123;
                b[p] = a[i];
                <span class="hljs-keyword">break</span>;
            &#125; <span class="hljs-keyword">else</span> &#123;
                <span class="hljs-keyword">if</span>(p == <span class="hljs-number">0</span>) <span class="hljs-keyword">break</span>;
                a[i] ^= b[p];
            &#125;
        &#125; 
    &#125;
    ll ans = <span class="hljs-number">0</span>;
    per(i,<span class="hljs-number">55</span>,<span class="hljs-number">0</span>)  ans = ans &gt; (ans ^ b[i]) ? ans : (ans ^ b[i]);
    <span class="hljs-built_in">cout</span> &lt;&lt; ans &lt;&lt; <span class="hljs-string">&quot;\n&quot;</span>;
    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
&#125;</code></pre></div>



    </div>
</div>
<p><strong><a href="P3857 [TJOI2008]彩灯">P3857 [TJOI2008]彩灯</a></strong></p>
<p>这道题也是一道板子题。我们先二进制状压一下每个开关的效果，然后求这些数的线性基。由于线性基的性质，所以可以得到答案就是$2^{cnt}$，其中$cnt$表示线性基中的元素个数。</p>
<div class='spoiler collapsed'>
    <div class='spoiler-title'>
        Code
    </div>
    <div class='spoiler-content'>
        <div class="hljs"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;map&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;set&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cmath&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;queue&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;bitset&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;vector&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdio&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstring&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdlib&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;iostream&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;algorithm&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> rep(i,a,b) for(register int i = (a);i &lt;= (b);++i)</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> per(i,a,b) for(register int i = (a);i &gt;= (b);--i) </span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> mkp std::make_pair</span>
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ll;
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">unsigned</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ull;
<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">string</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cin</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cout</span>;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x,<span class="hljs-keyword">int</span> y)</span></span>&#123;<span class="hljs-keyword">return</span> x &lt; y;&#125;

<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> N = <span class="hljs-number">1e6</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> inf = <span class="hljs-number">1e9</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">double</span> eps = <span class="hljs-number">1e-7</span>;
<span class="hljs-keyword">int</span> _,n,m;
ll a[<span class="hljs-number">2</span>*N],b[<span class="hljs-number">2</span>*N],ans;
<span class="hljs-keyword">char</span> ch[<span class="hljs-number">100</span>];

<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">int</span> <span class="hljs-title">find</span><span class="hljs-params">(ll x)</span></span>&#123;
    <span class="hljs-keyword">int</span> p = <span class="hljs-number">55</span>;
    <span class="hljs-keyword">while</span>(!(x &amp; (<span class="hljs-number">1l</span>l&lt;&lt;p)) &amp;&amp; p &gt;= <span class="hljs-number">1</span>) --p;
    <span class="hljs-keyword">return</span> p;
&#125;  

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    <span class="hljs-built_in">std</span>::ios::sync_with_stdio(<span class="hljs-number">0</span>);<span class="hljs-built_in">cin</span>.tie(<span class="hljs-number">0</span>);<span class="hljs-built_in">cout</span>.tie(<span class="hljs-number">0</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">ifdef</span> LOCAL <span class="hljs-comment">//use &quot;g++ xxx.cpp -DLOCAL&quot;</span></span>
        freopen(<span class="hljs-string">&quot;in.in&quot;</span>, <span class="hljs-string">&quot;r&quot;</span>, <span class="hljs-built_in">stdin</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">endif</span></span>
    <span class="hljs-built_in">cin</span> &gt;&gt; n &gt;&gt; m;
    rep(i,<span class="hljs-number">1</span>,m)&#123;
        <span class="hljs-built_in">cin</span> &gt;&gt; ch+<span class="hljs-number">1</span>;
        rep(j,<span class="hljs-number">1</span>,n) a[i] = (a[i] &lt;&lt; <span class="hljs-number">1</span>) + (ch[j] == <span class="hljs-string">&#x27;O&#x27;</span>);
    &#125;
    rep(i,<span class="hljs-number">1</span>,m)&#123;
        <span class="hljs-keyword">int</span> p;
        <span class="hljs-keyword">while</span>(<span class="hljs-number">1</span>)&#123;
            p = find(a[i]);
            <span class="hljs-keyword">if</span>(!b[p])&#123;
                b[p] = a[i];
                <span class="hljs-keyword">break</span>;
            &#125; <span class="hljs-keyword">else</span> &#123;
                a[i] ^= b[p];
                <span class="hljs-keyword">if</span>(p == <span class="hljs-number">0</span>) <span class="hljs-keyword">break</span>;
                <span class="hljs-keyword">continue</span>;
            &#125;
        &#125;
    &#125;
    ans = <span class="hljs-number">1</span>;
    per(i,<span class="hljs-number">55</span>,<span class="hljs-number">0</span>) <span class="hljs-keyword">if</span>(b[i])&#123;ans &lt;&lt;= <span class="hljs-number">1</span> , ans %= <span class="hljs-number">2008</span>;&#125;
    <span class="hljs-keyword">if</span>(ans == <span class="hljs-number">1</span>) ans = <span class="hljs-number">0</span>;
    <span class="hljs-built_in">cout</span> &lt;&lt; ans%<span class="hljs-number">2008</span> &lt;&lt; <span class="hljs-string">&quot;\n&quot;</span>;
    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
&#125;</code></pre></div>

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